Engaging in counterarguments (CAs) has been shown to be a promising approach to effectively support the development of critical thinking skills in educational settings. Developing an argumentation tutoring system that automatically provides personalized feedback on CAs would benefit both teachers and students by mitigating the constraints imposed by limited instructional resources. However, the development of such systems remains underexplored. To address this gap, we develop CAT, a CounterArgument Tutoring system that analyzes the underlying logical structure of CAs and provides tailored feedback accordingly. To evaluate CAT, we conduct a user study with 39 undergraduate students over two weeks in a real classroom, comparing it against an existing argumentation tutoring system as a baseline. The results indicate that CAT supports students in critically evaluating opposing arguments, considering alternative perspectives, and refining their own reasoning. Statistical analysis further shows that CAT elicits more engaged interaction than the baseline system. Moreover, students perceive CAT as easy to use and valuable for improving both their critical thinking and argumentation skills.